Nil 3-Armendariz Rings

Affiliation(s)

1Department of Mathematics, Northwest Normal University, Lanzhou, China 2Department of Mathematics, Khartoum University, Omdurman, Sudan.

Department of Mathematics, Northwest Normal University, Lanzhou, China.

1Department of Mathematics, Northwest Normal University, Lanzhou, China 2Department of Mathematics, Khartoum University, Omdurman, Sudan.

Department of Mathematics, Northwest Normal University, Lanzhou, China.

ABSTRACT

We introduce nil 3-Armendariz rings, which are generalization of 3-Armendariz rings and nil Armendaiz rings and investigate their properties. We show that a ring *R* is nil 3-Armendariz ring if and only if for any , *T*_{n}(*R*) is nil 3-Armendariz ring. Also we prove that a right Ore ring *R* is nil 3-Armendariz if and only if so is *Q*, where *Q* is the classical right quotient ring of *R*. With the help of this result, we can show that a commutative ring *R* is nil 3-Armendariz if and only if the total quotient ring of *R* is nil 3-Armendariz.

Cite this paper

E. Ali, A. Elshokry and Z. Liu, "Nil 3-Armendariz Rings,"*Advances in Pure Mathematics*, Vol. 3 No. 9, 2013, pp. 703-708. doi: 10.4236/apm.2013.39096.

E. Ali, A. Elshokry and Z. Liu, "Nil 3-Armendariz Rings,"

References

[1] E. P. Armendariz, “A Note on Extensions of Baer and p.p.-Rings,” Journal of the Australian Mathematical Society, Vol. 18, No. 4, 1974, pp. 470-473.

http://dx.doi.org/10.1017/S1446788700029190

[2] M. B. Rege and S. Chhawchharia, “Armendariz Rings,” Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 1, 1997, pp. 14-17.

http://dx.doi.org/10.3792/pjaa.73.14

[3] D. D. Anderson and V. Camillo, “Armendariz Rings and Gaussian Rings,” Communications in Algebra, Vol. 26, No. 7, 1998, pp. 2265-2272.

http://dx.doi.org/10.1080/00927879808826274

[4] N. K. Kim and Y. Lee, “Armendariz Rings and Reduced Rings,” Journal of Algebra, Vol. 223, No. 2, 2000, pp. 477-488. http://dx.doi.org/10.1006/jabr.1999.8017

[5] C. Huh, Y. Lee and A. Smoktunowicz, “Armendariz Rings and Semicommutative Rings,” Communications in Algebra, Vol. 30, No. 2, 2002, pp. 751-761.

http://dx.doi.org/10.1006/jabr.1999.8017

[6] T. K. Lee and T. L. Wong, “On Armendariz Rings,” Houston Journal of Mathematics, Vol. 29, No. 3, 2003, pp. 583-593.

[7] Y. Suiyi, “On the Extension of Armendariz Rings,” MS Thesis, Lanzhou University, Lanzhou, 2008.

[8] R. Antoine, “Nilpotent Elements and Armendariz Rings,” Journal of Algebra, Vol. 319, No. 8, 2008, pp. 3128-3140.

http://dx.doi.org/10.1016/j.jalgebra.2008.01.019

[9] Z. K. Liu and R. Y. Zhao, “On Weak Armendariz Rings,” Communications in Algebra, Vol. 34, No. 7, 2006, pp. 2607-2616. http://dx.doi.org/10.1080/00927870600651398

[10] H. F. Wu, “On Weak 3-Armendariz Rings,” Journal of Hangzhou Normal University (Natural Science Edition), Vol. 11, No. 3, 2012, pp. 241-244.

[11] H. F. Wu, “Extensions of Reduced Rings,” Journal of Hangzhou Normal University (Natural Science Edition), Vol. 10, No. 5, 2011, pp. 407-410.

[12] J. C. McConnell and J. C. Robson, “Noncommutative Noetherian Rings,” A Wiley-Interscience Publication, John Wiley and Sons, Ltd., Chichester, 1987.

[1] E. P. Armendariz, “A Note on Extensions of Baer and p.p.-Rings,” Journal of the Australian Mathematical Society, Vol. 18, No. 4, 1974, pp. 470-473.

http://dx.doi.org/10.1017/S1446788700029190

[2] M. B. Rege and S. Chhawchharia, “Armendariz Rings,” Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 1, 1997, pp. 14-17.

http://dx.doi.org/10.3792/pjaa.73.14

[3] D. D. Anderson and V. Camillo, “Armendariz Rings and Gaussian Rings,” Communications in Algebra, Vol. 26, No. 7, 1998, pp. 2265-2272.

http://dx.doi.org/10.1080/00927879808826274

[4] N. K. Kim and Y. Lee, “Armendariz Rings and Reduced Rings,” Journal of Algebra, Vol. 223, No. 2, 2000, pp. 477-488. http://dx.doi.org/10.1006/jabr.1999.8017

[5] C. Huh, Y. Lee and A. Smoktunowicz, “Armendariz Rings and Semicommutative Rings,” Communications in Algebra, Vol. 30, No. 2, 2002, pp. 751-761.

http://dx.doi.org/10.1006/jabr.1999.8017

[6] T. K. Lee and T. L. Wong, “On Armendariz Rings,” Houston Journal of Mathematics, Vol. 29, No. 3, 2003, pp. 583-593.

[7] Y. Suiyi, “On the Extension of Armendariz Rings,” MS Thesis, Lanzhou University, Lanzhou, 2008.

[8] R. Antoine, “Nilpotent Elements and Armendariz Rings,” Journal of Algebra, Vol. 319, No. 8, 2008, pp. 3128-3140.

http://dx.doi.org/10.1016/j.jalgebra.2008.01.019

[9] Z. K. Liu and R. Y. Zhao, “On Weak Armendariz Rings,” Communications in Algebra, Vol. 34, No. 7, 2006, pp. 2607-2616. http://dx.doi.org/10.1080/00927870600651398

[10] H. F. Wu, “On Weak 3-Armendariz Rings,” Journal of Hangzhou Normal University (Natural Science Edition), Vol. 11, No. 3, 2012, pp. 241-244.

[11] H. F. Wu, “Extensions of Reduced Rings,” Journal of Hangzhou Normal University (Natural Science Edition), Vol. 10, No. 5, 2011, pp. 407-410.

[12] J. C. McConnell and J. C. Robson, “Noncommutative Noetherian Rings,” A Wiley-Interscience Publication, John Wiley and Sons, Ltd., Chichester, 1987.